The open box problem

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The open box problem

Introduction

The aim of my algebraic investigation into the open box problem is to determine the size of square cut which makes the volume of the box as large as possible for any given rectangular sheet of card. The problem itself is simple, an open box is made from a sheet of card, identical squares are then cut off each of the four corners, the sheet is then folded to make box. It is my aim to find out the maximum square cut which gives me the maximum volume box.

                                         

Strategy

  1. Try to find the size of cut-out that will give me the maximum volume of a piece of card 6cm x 6cm, progressing onto algebra.
  2. Look for limitations in the results.
  3. Devise a plan for overcoming these limitations.
  4. Extend the work to find a general formula that will help me to work with all squares and rectangles.

Technique

In this situation there is no need for us to go even close to a pair of scissors or piece of card, for this investigation I am to use Excel. Excel is a computer programme in which I can input information; it will then calculate this information and give me results for what different size cut outs for different sized card.

The set-up of this is quite simple;

Size of card is out into cell c3 in this case

                                        Volume is calculated

                                          With a real box by

Size of cut-out in                                Width x Depth x Height

Cells a6, a7 etc…                                The same happens here

                                                Corresponding cell              

                                                Numbers put in with *                                                         between each one                                                                 meaning multiply.

Width is calculated using formula.                

$ is used to keep the term c3                         

(the card size) constant, stops

it from being changed. It then minus’

2 x the cut-out giving length of width.

The same formula is then used for calculation

of depth.

Note: When putting formulas in for real you always put an = sign before typing a formula, Excel then knows your typing in a formula and will work out the answer to it when finished. Not done on above as wanted to show what the formulas were.

Results and Evidence

These are the results for a 6cm x 6cm piece of card.

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You can see in each case that a 1.00 cm size cut-out gives the biggest volume of the open box. At first I used whole integers as my cut-out size, I then wanted to see if it was a smaller number that would give me my biggest volume, so I increased my sample size. Again it is easy to see that 1.00 cm still gives the biggest volume, but I wanted to go further, maybe it was a number like 0.95 or 1.05 and the computer was rounding up/down so I limited my sample size again. But still 1.00 ...

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